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June  2012, 32(6): 2285-2299. doi: 10.3934/dcds.2012.32.2285

Heat Kernel estimates for some elliptic operators with unbounded diffusion coefficients

Giorgio Metafune 1, and  Chiara Spina 1,

1. 

Dipartimento di Matematica “Ennio De Giorgi”, Università del Salento, C.P.193, 73100, Lecce, Italy, Italy

Received  January 2011 Revised  July 2011 Published  February 2012

We prove heat kernel bounds for the operator $(1+|x|^\alpha)\Delta$ in $\mathbb{R}^N$, through Nash inequalities and weighted Hardy inequalities.
Keywords: Elliptic operators, Nash inequality, kernel estimates..
Mathematics Subject Classification: 47D07, 35B50, 35J25, 35J7.
Citation: Giorgio Metafune, Chiara Spina. Heat Kernel estimates for some elliptic operators with unbounded diffusion coefficients. Discrete & Continuous Dynamical Systems, 2012, 32 (6) : 2285-2299. doi: 10.3934/dcds.2012.32.2285
References:
[1]

A. Bazan and W. Neves, The Caffarelli-Kohn-Niremberg's inequality for arbitrary norms,, preprint., ().   Google Scholar

[2]

A. Bazan and W. Neves, The Hardy and Caffarelli-Kohn-Niremberg inequalities revised,, preprint, ().   Google Scholar

[3]

D. Bakry, F. Bolley, I. Gentil and P. Maheux, Weighted Nash inequalities,, preprint, ().   Google Scholar

[4]

E. B. Davies, "One-Parameter Semigroups," London Mathematical Society Monographs, 15, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London-New York, 1980.  Google Scholar

[5]

E. B. Davies, "Heat Kernels and Spectral Theory," Cambridge Tracts in Mathematics, 92, Cambridge University Press, Cambridge, 1989.  Google Scholar

[6]

K.-J. Engel and R. Nagel, "One-Parameter Semigroups for Linear Evolutions Equations," Graduate Texts in Mathematics, 194, Springer-Verlag, New York, 2000.  Google Scholar

[7]

S. Fornaro and L. Lorenzi, Generation results for elliptic operators with unbounded diffusion coefficients in $L^p$- and $C_b$-spaces, Discrete and Continuous Dynamical Systems, 18 (2007), 747-772.  Google Scholar

[8]

G. Metafune, E. M. Ouhabaz and D. Pallara, Long time behavior of heat kernels of operators with unbounded drift terms, J. Math. Anal. Appl., 377 (2011), 170-179. doi: 10.1016/j.jmaa.2010.10.023.  Google Scholar

[9]

G. Metafune and C. Spina, Elliptic operators with unbounded diffusion coefficients in $L^p$ spaces,, Ann. Sc. Norm. Super. Pisa Cl. Sci., ().   Google Scholar

[10]

G. Metafune and C. Spina, Kernel estimates for a class of Schrödinger semigroups, Journal of Evolution Equations, 7 (2007), 719-742. doi: 10.1007/s00028-007-0338-3.  Google Scholar

[11]

G. Metafune, D. Pallara and M. Wacker, Feller semigroups on $\R^N$, Semigroup Forum, 65 (2002), 159-205. doi: 10.1007/s002330010129.  Google Scholar

[12]

B. Muckenhoupt, Hardy's inequalities with weights, Studia Math., 44 (1972), 31-38.  Google Scholar

[13]

B. Muckenhoupt and R. Wheeden, Weighted norm inequalities for fractional integrals, Trans. Amer. Math. Soc., 192 (1974), 261-274. doi: 10.1090/S0002-9947-1974-0340523-6.  Google Scholar

[14]

E. M. Ouhabaz, "Analysis of Heat Equations on Domains," London Mathematical Society Monographs Series, 31, Princeton University Press, Princeton, NJ, 2005.  Google Scholar

[15]

F.-Y. Wang, Functional inequalities and spectrum estimates: The infinite measure case, J. Funct. Anal., 194 (2002), 288-310. doi: 10.1006/jfan.2002.3968.  Google Scholar

show all references

References:
[1]

A. Bazan and W. Neves, The Caffarelli-Kohn-Niremberg's inequality for arbitrary norms,, preprint., ().   Google Scholar

[2]

A. Bazan and W. Neves, The Hardy and Caffarelli-Kohn-Niremberg inequalities revised,, preprint, ().   Google Scholar

[3]

D. Bakry, F. Bolley, I. Gentil and P. Maheux, Weighted Nash inequalities,, preprint, ().   Google Scholar

[4]

E. B. Davies, "One-Parameter Semigroups," London Mathematical Society Monographs, 15, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London-New York, 1980.  Google Scholar

[5]

E. B. Davies, "Heat Kernels and Spectral Theory," Cambridge Tracts in Mathematics, 92, Cambridge University Press, Cambridge, 1989.  Google Scholar

[6]

K.-J. Engel and R. Nagel, "One-Parameter Semigroups for Linear Evolutions Equations," Graduate Texts in Mathematics, 194, Springer-Verlag, New York, 2000.  Google Scholar

[7]

S. Fornaro and L. Lorenzi, Generation results for elliptic operators with unbounded diffusion coefficients in $L^p$- and $C_b$-spaces, Discrete and Continuous Dynamical Systems, 18 (2007), 747-772.  Google Scholar

[8]

G. Metafune, E. M. Ouhabaz and D. Pallara, Long time behavior of heat kernels of operators with unbounded drift terms, J. Math. Anal. Appl., 377 (2011), 170-179. doi: 10.1016/j.jmaa.2010.10.023.  Google Scholar

[9]

G. Metafune and C. Spina, Elliptic operators with unbounded diffusion coefficients in $L^p$ spaces,, Ann. Sc. Norm. Super. Pisa Cl. Sci., ().   Google Scholar

[10]

G. Metafune and C. Spina, Kernel estimates for a class of Schrödinger semigroups, Journal of Evolution Equations, 7 (2007), 719-742. doi: 10.1007/s00028-007-0338-3.  Google Scholar

[11]

G. Metafune, D. Pallara and M. Wacker, Feller semigroups on $\R^N$, Semigroup Forum, 65 (2002), 159-205. doi: 10.1007/s002330010129.  Google Scholar

[12]

B. Muckenhoupt, Hardy's inequalities with weights, Studia Math., 44 (1972), 31-38.  Google Scholar

[13]

B. Muckenhoupt and R. Wheeden, Weighted norm inequalities for fractional integrals, Trans. Amer. Math. Soc., 192 (1974), 261-274. doi: 10.1090/S0002-9947-1974-0340523-6.  Google Scholar

[14]

E. M. Ouhabaz, "Analysis of Heat Equations on Domains," London Mathematical Society Monographs Series, 31, Princeton University Press, Princeton, NJ, 2005.  Google Scholar

[15]

F.-Y. Wang, Functional inequalities and spectrum estimates: The infinite measure case, J. Funct. Anal., 194 (2002), 288-310. doi: 10.1006/jfan.2002.3968.  Google Scholar

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